Essay About Modular Arithmetics And Main Drawback

Modular Arithmetics
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MODULAR ARITHMETICS
Modular arithmetic can be used to compute
exactly, at low cost, a set of simple computations.
These include most geometric predicates, that
need to be checked exactly, and especially, the
sign of determinants and more general polynomial
expressions.
Modular arithmetic resides on the Chinese
Remainder Theorem, which states that, when
computing an integer expression, you only have to
compute it modulo several relatively prime integers
called the modulis. The true integer value can then
be deduced, but also only its sign, in a simple and
efficient maner.
The main drawback with modular arithmetic is its
static nature, because we need to have a bound on
the result to be sure that we preserve ourselves
from overflows (that cant be detected easily
while computing). The smaller this known bound is,
the less computations we have to do.
We have developed
a set of efficient tools to deal
with these problems, and we propose a filtered
approach, that is, an approximate computation
using floating point arithmetic, followed, in the bad
case, by a modular computation of the expression
of which we know a bound, thanks to the floating
point computation we have just done. Theoretical